50,000-4,000/5coins= 9200 for 1 return
prob=2.08125E-05/60=3.46875E-07
er=3.46875E-07x9200=0.003191251
total er=1.001725232+0.003191251=1.004916483
var=3.46875E-07(9200-1.004916483)^2=29.35309465
total var=28.25553947+29.35309465=57.60863412
bankroll(11%ror)=var/(er-1)=11717.44762 bets
N0(std.dev.=er)=var/(er-1)^2=2,383,298.631 hands
bankroll growth index=10^7x(er-1)^2/var=4.195865289
10/7db+50,000seqroyal
--- In vpFREE@yahoogroups.com, "nightoftheiguana2000"
<nightoftheiguana2000@y...> wrote:
50,000-4,000/5coins= 9200 for 1 return
prob=2.08125E-05/60=3.46875E-07
er=3.46875E-07x9200=0.003191251
total er=1.001725232+0.003191251=1.004916483
var=3.46875E-07(9200-1.004916483)^2=29.35309465
total var=28.25553947+29.35309465=57.60863412
bankroll(11%ror)=var/(er-1)=11717.44762 bets
N0(std.dev.=er)=var/(er-1)^2=2,383,298.631 hands
bankroll growth index=10^7x(er-1)^2/var=4.195865289
*** var=3.46875E-07(9200-1.004916483)^2=29.35309465 ***
*** total var=28.25553947+29.35309465=57.60863412 ***
How come you're using the overall game average here rather than just
the excess value on the seq royal? Seems like the rest of the game
return is already included in the variance on the game, the 28.2555
piece, and that this line should read
### var=3.46875E-07(9200-0.003191251)^2=29.35947963 ###
(Not that a difference in the 5th significant digit matters overall,
but what good are math skills if you can't argue over the 5th decimal
place once in a while, LOL. My other objection is more substantial,
however.)
And then from here, I don't think you can simply add variances of two
pieces to get the variance overall. It made sense at first, but I
tried some examples and they didn't work. For example, the series
(0,1,3,3,8) has a var of 7.6 and (1,2,3,4,5) of 2.0, but the sum of
the two (1,3,6,7,13) has a var of 16.8. The difference would be even
greater if the two series were complimentary (one filled in the
swings of the other) although that, of course, in not the case with
the seq royal, so I didn't bother with an example like that.
On the general topic of seq royals, we just got two new banks of
games in tunica with progressive seq royals. One of them is already
up to 37000:1 excess return on the seq (versus 9200:1 in your
example). That game hasn't broke positive yet due to a lousy
underlying game, and when/if it does the seq royal will be like 2-3%
of the overall return. On a game like that, even with an ER over
1.00, can you really call it a positive expectation game when the
normalization cycle is a billion hands? Let's see, at 600 hph, one
billion hands would take about 190 years nonstop to play without
even a bathroom break.
How come you're using the overall game average here rather than
just
the excess value on the seq royal? Seems like the rest of the game
return is already included in the variance on the game, the 28.2555
piece, and that this line should read
### var=3.46875E-07(9200-0.003191251)^2=29.35947963 ###
I disagree. Variance is a measure of the distance from the total ER.
And then from here, I don't think you can simply add variances of
two
pieces to get the variance overall.
You can if you're careful about it. I wasn't that careful so let me
take another stab at it:
old royal variance: 2.08125E-05(800-1.001725232)^2=
13.28666343
new seq. royal variance: (2.08125E-05/60)(10,000-1.004916483)^2=
34.68052874
new non-seq. royal variance: (59/60)(2.08125E-05)(800-1.004916483)^2=
13.06511468
new total variance: 28.25553947-13.28666343+34.68052874+13.06511468=
62.71451946
You should also make a few strategy variations, but since they happen
so rarely I don't think they effect the numbers that much.
On the general topic of seq royals, we just got two new banks of
games in tunica with progressive seq royals. One of them is already
up to 37000:1 excess return on the seq (versus 9200:1 in your
example). That game hasn't broke positive yet due to a lousy
underlying game, and when/if it does the seq royal will be like 2-
3%
of the overall return. On a game like that, even with an ER over
1.00, can you really call it a positive expectation game when the
normalization cycle is a billion hands? Let's see, at 600 hph, one
billion hands would take about 190 years nonstop to play
without
even a bathroom break.
It's positive, but not all positive plays are good plays. That's why
you want to know what N0 (a normalization cycle) is. Here's how I
approach any gamble: Do I know the strategy? What is the er of that
strategy and is it positive? What is the variance and do I have the
bankroll to play? What is the N0? What is the Bankroll Growth Index
and how does it compare to other gambles I might spend time on?
···
--- In vpFREE@yahoogroups.com, "blaw57" <blaw57@y...> wrote:
--- In vpFREE@yahoogroups.com, "nightoftheiguana2000"
<nightoftheiguana2000@y...> wrote:
You can if you're careful about it. I wasn't that careful so let me
take another stab at it:
old royal variance: 2.08125E-05(800-1.001725232)^2=
13.28666343
new seq. royal variance: (2.08125E-05/60)(10,000-1.004916483)^2=
34.68052874
new non-seq. royal variance: (59/60)(2.08125E-05)(800-1.004916483)
^2=
13.06511468
new total variance: 28.25553947-13.28666343+34.68052874+13.06511468=
62.71451946
Closer, but still not exact, I don't think.
Variance = the sum of the squared difference-from-average (ER) of
every hand divided by the number of hands
Your adjustment for the royals is now exact but you still have to
recalculate the non-royal portion of the original 28.2555 variance
because the ER has changed, and the original ER was part of the
difference from every hand. We know that number will now be lower,
but I don't see any way to calculate exactly how much lower without
redoing the entire hand-by-hand calculation.
It's positive, but not all positive plays are good plays. That's
why
you want to know what N0 (a normalization cycle) is. Here's how I
approach any gamble: Do I know the strategy? What is the er of that
strategy and is it positive? What is the variance and do I have the
bankroll to play? What is the N0? What is the Bankroll Growth Index
and how does it compare to other gambles I might spend time on?
I generally agree with you on this, but to quote (or at least
paraphrase) you, positive gambling is still gambling, and this one
seems more of a gamble than most. If I go 100,000 hands without a
royal, I can comfort myself with the thought that things are likely
to even out in the long run. If I go 100,000 without a SR, there is
no such comfort, as I won't be alive for the long run in that game.
And anyhow, again to paraphrase you, positive gambling may still be
gambling, but it beats the pants off negative gambling. I would a lot
sooner play this game than I would drop a coin the wheel of fortune
machine (what, somebody has a picture? that can't be me, I wouldn't
do that - sure does look like me, though - well, maybe I was on drugs
that day or something).
Agreed, the effect is small however. Instead of 28.25553947 I get
28.25554966
···
--- In vpFREE@yahoogroups.com, "blaw57" <blaw57@y...> wrote:
Closer, but still not exact, I don't think.
Variance = the sum of the squared difference-from-average (ER) of
every hand divided by the number of handsYour adjustment for the royals is now exact but you still have to
recalculate the non-royal portion of the original 28.2555 variance
because the ER has changed, and the original ER was part of the
difference from every hand. We know that number will now be lower,
but I don't see any way to calculate exactly how much lower without
redoing the entire hand-by-hand calculation.
--- In vpFREE@yahoogroups.com, "nightoftheiguana2000"
<nightoftheiguana2000@y...> wrote:
Agreed, the effect is small however. Instead of 28.25553947 I get
28.25554966
Ha! A win at 5th decimal place! 25 geek points for me!
Actually, I thought it was informative to get into the nitty gritty
of the variance calculation. I hope a few other readers benefitted as
well. Back to my original question/observation, because the variance
calculation is based upon squares of differences, a few large
differences are going to have a disproportionate impact, and a single
large term that gets larger will have a major impact. 800 (a royal)
squared is 640,000, which over 40,000 hands (the royal cycle) adds 16
to the variance. Take JOB, with its 19.5 variance, and remove the
royal - the rest of the game has roughly a 3.5 variance. Now, double
the royal as in a progressive game, and now it adds 4*16 or 64 to the
variance, or roughly 67.5 for the game overall. Nothing about the
game has changed except the royal, which you almost never
experieince, and yet variance is all over the lot from 3 to 67.
Now, the pertinent question is, is that what you want to see in a
risk/volatility measurement for VP? I remember back in Stat class
wondering why they chose squaring as a means of emphasizing
differences in Std Dev - why not just double the difference or
quadruple the difference? You would still have a measure that
distinguishes between low-volatility patterns and high-volatility
patterns with the same average, but now the relationship to
volatility would be a straight line instead of an ascending curve.
Well, there are actually answers to those questions, and that's why
SD and Variance have hung around as statistical measures. But whether
those reasons prove that variance is the most useful single index
number for risk/volatility in VP games is another matter.